At room temperature (20°C), a 5.0-cm-long brass rod is 20 μm too long to fit into a slot. To what temperature should you cool the rod so that it just barely fits?

검증된 단계별 안내

Step 1: Understand the problem. The brass rod is currently too long to fit into the slot by 20 μm. To solve this, we need to calculate the temperature at which the rod contracts enough to fit. This involves the concept of thermal expansion, where the length of a material changes with temperature.

Step 2: Write the formula for linear thermal expansion:

ΔL = L₀αΔT

, where ΔL is the change in length, L₀ is the initial length of the rod, α is the coefficient of linear expansion for brass, and ΔT is the change in temperature.

Step 3: Rearrange the formula to solve for ΔT:

ΔT = ΔL / (L₀α)

. Here, ΔL is the amount the rod needs to contract (20 μm = 20 × 10⁻⁶ m), L₀ is the initial length of the rod (5.0 cm = 0.050 m), and α is the coefficient of linear expansion for brass (approximately 19 × 10⁻⁶ /°C).

Step 4: Substitute the known values into the formula:

ΔT = (20 × 10⁻⁶) / (0.050 × 19 × 10⁻⁶)

. This will give the temperature change required for the rod to contract by 20 μm.

Step 5: Determine the final temperature. Since the initial temperature is 20°C, subtract the calculated ΔT from 20°C to find the temperature to which the rod must be cooled.

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주요 개념

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Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size or volume in response to changes in temperature. For solids, this is typically linear expansion, where the length of an object increases with temperature. The relationship is often described by the formula ΔL = αL₀ΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the change in temperature.

Coefficient of Linear Expansion

The coefficient of linear expansion (α) is a material-specific constant that quantifies how much a material expands per degree of temperature change. For brass, this value is approximately 19 x 10⁻⁶ /°C. Understanding this coefficient is crucial for calculating how much a given length of brass will contract or expand when subjected to temperature changes.

Temperature Change Calculation

To determine the necessary temperature change to achieve a specific length adjustment, one can rearrange the thermal expansion formula. By knowing the initial length, the desired length, and the coefficient of linear expansion, one can solve for the change in temperature (ΔT). This calculation is essential for finding the temperature at which the brass rod will fit into the slot.